$p$-adic hypergeometric function related with $p$-adic multiple polylogarithms

This paper introduces a $p$-adic analogue of Gauss’s hypergeometric function, constructed via a method that is distinct from distinct from Dwork’s approach. The idea of our construction is motivated by the Ohno-Zagier formula, which is elucidated through the relationship between the hypergeometric differential equation and the Knizhnik-Zamolodchikov (KZ) equation. We develop a rigorous framework for the residue-wise analytic prolongation of our $p$-adic hypergeometric function by exploring its relationship with $p$-adic multiple polylogarithms. Through a detailed analysis of its local behavior near the point $1$, we show a $p$-adic version of Gauss hypergeometric theorem for the function.